Group Theory (LXV): Let G be a group, H a subgroup of G, T an automorphism of G. Let (H)T = {hT| hєH}.Prove (H)T is a subgroup of G.

Modern Algebra Group Theory (LXV) Automorphism of a Group Let G be a group, H a subgroup of G, T an automorphism of G. Let (H)T = {hT| hєH} ...continues

Group Theory (LXVI): Let G be a group, T an automorphism of G, N a normal subgroup of G. Prove that (N)T is a normal subgroup of G.

Modern Algebra Group Theory (LXVI) Automorphism of a Group Let G be a group, T an automorphism of G, N a normal subgroup of G. Prove that (N)T is a normal subgroup of G.

Group Theory (LXVII): Prove that I(G), the set of all inner automorphisms of G is a subgroup of Aut(G), the set all automorphisms of G.

Modern Algebra Group Theory (LXVII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group Prove that I(G), the set of all inner automorphisms of G is a subgroup of Aut(G), the set all automorphisms of G.

Group Theory (LXVIII): Prove that I(G), the set of all inner automorphisms of G is a group.

Modern Algebra Group Theory (LXVIII) The Set of all Inner Automorphisms of a Group Prove that I(G), the set of all inner automorphisms of G is a group.

Group Theory (LXIX): Prove that I(G), the set of all inner automorphisms of G is normal in Aut(G), the set of all automorphisms of G.

Modern Algebra Group Theory (LXIX) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group Prove that I(G), the ...continues

Group Theory (LXX): If G is a group, then Aut(G), the set of automorphisms of G, is also a group.

Modern Algebra Group Theory (LXX) The Set of all Automorphisms of a Group If G is a group, then Aut(G), the set of automorphisms of G, is also a group.

Group Theory (LXXI): Let φ be a homomorphism of G onto G¯ with kernel K . Then G/K ≈ G¯

Modern Algebra Group Theory (LXXI) Homomorphism of a Group The Kernel of a Homomorphism ...continues

Group Theory (LXXII): Let G be a group and Z(G), the centre of G, then G/Z(G) ≈ I(G) , where I(G) is the set of all inner automorphisms of G.

Modern Algebra Group Theory (LXXII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group Let G be a group and Z(G), the centre of G, then G/Z(G)  ...continues

Group Theory (LXXIII): Let G be a group, consider the mapping of G into itself, λg, defined for gЄG by x λg =gx for all xЄG.Prove that λg is one-to-one and onto, and that λgh = λhλg.

Modern Algebra Group Theory (LXXIII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group Let G be a group, consider the mapping of G into itself, ...continues

Group Theory (LXXIV): Let G be a group, consider the mapping λg :G→G defined for gЄG by xλg = gx for all xЄG and the mapping τg :G→G defined for xЄG by xτg = xg for every xЄG. Prove that for any g,hЄG, the mappings λg , τh satisfy λg τh = τh λg .

Modern Algebra Group Theory (LXXIV) The Set of all Automorphisms of a Groupn The Set of all Inner Automorphisms of a Group ...continues